Dunford–Schwartz theorem

In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz states that the averages of powers of certain norm-bounded operators on L¹ converge in a suitable sense.

Theorem. Let $T$ be a linear operator from $L_{1}$ to $L_{1}$ with $\|T\|_{1}\leq 1$ and $\|T\|_{\infty }\leq 1$. Then

$\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}T^{k}f$

exists almost everywhere for all $f\in L_{1}$.

The statement is no longer true when the boundedness condition is relaxed to even $\|T\|_{\infty }\leq 1+\varepsilon$.